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Tamagawa Number Conjecture for Zeta Values 165 ICM 2002 · Vol. III · 1–3 Tamagawa Number Conjecture for zeta Values Kazuya Kato* Abstract Spencer Bloch and the author formulated a general conjecture (Tama- gawa number conjecture) on the relation between values of zeta functions of motives and arithmetic groups associated to motives. We discuss this conjec- ture, and describe some application of the philosophy of the conjecture to the study of elliptic curves. 2000 Mathematics Subject Classification: 11G40. Keywords and Phrases: zeta function, Etale cohomology, Birch Swinnerton- Dyer conjecture. Mysterious relations between zeta functions and various arithmetic groups have been important subjects in number theory. (0.0) zeta functions ↔ arithmetic groups. A classical result on such relation is the class number formula discovered in 19th century, which relates zeta functions of number field to ideal class groups and unit groups. As indicated in (0.1)–(0.3) below, the formula of Grothendieck ex- pressing the zeta functions of varieties over finite fields by etale cohomology groups, Iwasawa main conjecture proved by Mazur-Wiles, and Birch and Swinnerton-Dyer conjectures for abelian varieties over number fields, considered in 20th century, also have the form (0.0). (0.1) Formula of Grothendieck. zeta functions ↔ etale cohomology groups. (0.2) Iwasawa main conjecture. zeta functions, zeta elements ↔ ideal class groups, unit groups. (0.3) Birch Swinnerton-Dyer conjectures (see 4). arXiv:math/0304233v1 [math.NT] 16 Apr 2003 zeta functions ↔ groups of rational points, Tate-Shafarevich groups. Here in (0.2), “zeta elements” mean cyclotomic units which are units in cyclo- tomic fields and closely related to zeta functions. Roughly speaking, the relations *Department of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro, Tokyo, Japan. E-mail: [email protected] 164 KazuyaKato (often conjectural) say that the order of zero or pole of the zeta function at an in- teger point is equal to the rank of the related finitely generated arithmetic abelian group (Tate, the conjecture (0.3), Beilinson, Bloch, ...) and the value of the zeta function at an integer point is related to the order of the related arithmetic finite group. In [BK], Bloch and the author formulated a general conjecture on (0.0) (Tama- gawa number conjecture for motives). Further generalizations of Tamagawa number conjecture by Fontaine, Perrin-Riou, and the author [FP], [Pe1] [Ka1], [Ka2] have the form (0.4) zeta functions (= Euler products, analytic) ↔ zeta elements (= Euler systems, arithmetic) ↔ arithmetic groups. Here the first ↔ means that zeta functions enter the arithmetic world transforming themselves into zeta elements, and the second ↔ means that zeta elements generate “determinants” of certain etale cohomology groups. The aim of this paper is to discuss (0.4) in an expository style. We review (0.1) in §1, and then in §2, we describe the generalized Tamagawa number conjecture (0.4), the relation with (0.2), and an application of the philosophy (0.4) to (0.3). In this paper, we fix a prime number p. For a commutative ring R, let Q(R) be the total quotient ring of R obtained from R by inverting all non-zerodivisors. 1. Grothendieck formula and zeta elements Let X be a scheme of finite type over a finite field Fq. We assume p is different from char(Fq). In this §1, we first review the formula (1.1.2) of Grothendieck representing zeta functions of p-adic sheaves on X by etale cohomology. We then show that those zeta functions are recovered from p-adic zeta elements (1.3.5). 1.1. Zeta functions and etale cohomology groups in positive charac- teristic case. −s −1 The Hasse zeta function ζ(X,s) = Qx∈|X| (1 − ♯κ(x) ) , where |X| denotes the set of all closed points of x and κ(x) denotes the residue field of x, −s has the form ζ(X,s)= ζ(X/Fq, q ) where deg(x) −1 ζ(X/Fq,u)= Y (1 − u ) , deg(x) = [κ(x): Fq]. (1.1.1) x∈|X| A part of Weil conjectures was that ζ(X/Fq,u) is a rational function in u, and it was proved by Dwork and then slighly later by Grothendieck. The proof of Grothendieck gives a presentation of ζ(X/Fq,u) by using etale cohomologyy. More generally, for a finite extension L of Qp and for a constructible L-sheaf F on X, Grothendieck proved that the L-function L(X/Fq, F,u) has the presentation −1 m ¯ (−1)m L(X/Fq, F,u)= Y detL(1 − ϕqu ; Het,c(X ⊗Fq Fq, F)) (1.1.2) m Tamagawa Number Conjecture for zeta Values 165 m where Het,c is the etale cohomology with compact supports and ϕq is the action of the q-th power morphism on X. In the case L = Qp = F, ζ(X/Fq,u)= L(X/Fq, F,u). 1.2. p-adic zeta elements in positive characteristic case. Determinants appear in the theory of zeta functions as above, rather often. The regulator of a number field, which appears in the class number formula, is a determinant. Such relation with determinant is well expressed by the notion of “determinant module”. If R is a field, for an R-module V of dimension r, detR(V ) means the 1 dimen- r sional R-module ∧R(V ). For a bounded complex C of R-modules whose cohomolo- m m ⊗(−1)m gies H (C) are finite dimensional, detR(C) means ⊗m∈Z {detR(H (C))} . This definition is generalized to the definition of an invertible R-module detR(C) associated to a perfect complex C of R-modules for a commutative ring R (see −1 [KM]). detR (C) means the inverse of the invertible module detR(C). By a pro-p ring, we mean a topological ring which is an inverse limit of finite rings whose orders are powers of p. Let Λ be a commutative pro-p ring. By a ctf Λ-complex on X, we mean a complex of Λ-sheaves on X for the etale topology with constructible cohomology sheaves and with perfect stalks. For a ctf Λ-complex F on X, RΓet,c(X, F) (c means with compact supports) is a perfect complex over Λ. For a commutative pro-p ring Λ and for a ctf Λ-complex F on X, we define the −1 p-adic zeta element ζ(X, F, Λ) which is a Λ-basis of detΛ RΓet,c(X, F). Consider the distinguished triangle ¯ ¯ RΓet,c(X, F) → RΓet,c(X ⊗Fq Fq, F)@ > 1 − ϕ>>RΓet,c(X ⊗Fq Fq, F). (1.2.1) Since det is multiplicative for distinguished triangles, (1.2.1) induces an isomorphism −1 −1 ∼ F ¯ F ¯ ∼ detΛ RΓet,c(X, F) = detΛ RΓet,c(X ⊗ q Fq, F) ⊗Λ detΛRΓet,c(X ⊗ q Fq, F) = Λ. (1.2.2) −1 We define ζ(X, F, Λ) to be the image of 1 ∈ Λ in detΛ RΓet,c(X, F) under (1.2.2). −1 It is a Λ-basis of the invertible Λ-module detΛ RΓet,c(X, F). 1.3. Zeta functions and p-adic zeta elements in positive character- istic case. Let L be a finite extension of Qp, let OL be the valuation ring of L, and let F be a constructible OL-sheaf on X. We show that the zeta function L(X/Fq, FL,u) of the L-sheaf FL = F ⊗OL L is recovered from a certain p-adic zeta element as in (1.3.5) below. Let ¯ Λ= OL[[Gal(Fq/Fq)]] =←− lim OL[Gal(Fqn /Fq)]. (1.3.1) n Let s(Λ) be the Λ-module Λ which is regarded as a sheaf on the etale site of X via the natural action of Gal(F¯ q/Fq). Then m m ∼ F F n Het,c(X, F ⊗OL s(Λ)) = ←−lim Het,c(X ⊗ q q , F) (1.3.2) n 166 KazuyaKato where the transition maps of the inverse system are the trace maps. From this, we m can deduce that Het,c(X, F⊗OL s(Λ)) is a finitely generated OL-module for any m. Hence we have Q(Λ)⊗Λ RΓet,c(X, F⊗OL s(Λ)) = 0 and this gives an identificatition canonical isomorphism −1 Q(Λ) ⊗Λ detΛ RΓet,c(X, F ⊗OL t(Λ)) = Q(Λ). (1.3.3) Note n Q(Λ) = Q(lim←− OL[u]/(u − 1)) ⊃ Q(OL[u]) = L(u). (1.3.4) n By a formal argument, we can prove the following (1.3.5) (1.3.6) which show zeta function = zeta element, zeta value = zeta element, respectively. L(X/Fq, FL,u)= ζ(X, F ⊗OL s(Λ), Λ) in Q(Λ). (1.3.5) m If Het,c(X, FL) =0 for any m, L(X/Fq, FL,u) has no zero or pole at u = 1, and L(X/Fq, FL, 1) = ζ(X, F, OL) in L. (1.3.6) 2. Tamagawa number conjecture In 2.1, we describe the generalized version of Tamagawa number conjecture. In 2.2 (resp. 2.3), we consider p-adic zeta elements associated to 1 (resp. 2) di- mensional p-adic representations of Gal(Q¯ /Q), and their relations to (0.2) (resp. (0.3)). 1 2.1. The conjecture. Let X be a scheme of finite type over Z[ p ]. For a complex of sheaves F on X for the etale topology, we define the compact support version RΓet,c(X, F) of RΓet(X, F) as the mapping fiber of 1 RΓet(Z[ ],Rf F) → RΓet(R,Rf F) ⊕ RΓet(Qp,Rf F).
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